Held group - Knowlpedia

The Held group $\mathrm{He}$ is the group with generators $a,b$ and the following complete defining relations:

\begin{gathered} a^2=b^7=(ab)^{17}=1,\\ [a,b]^6=[a,b^3]^5=1,\\ [a,babab^{-1}abab]=1,\\ (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2=1. \end{gathered}

Here $[x,y]=x^{-1}y^{-1}xy$ is the commutator. The presented group consists of words in $a,b$ and their inverses, identified by these relations; multiplication is concatenation followed by this identification.

Order and structure

This is a nonabelian finite simple group, with

|\mathrm{He}|=4030387200=2^{10}\cdot3^{3}\cdot5^{2}\cdot7^{3}\cdot17.

It belongs to the Monster display block of the sporadic groups. This grouping records the ATLAS organization, without asserting a direct embedding into another displayed group.

Reading the presentation

Every displayed equality is imposed on the free group on the two generators. This is a full presentation; the shorter generator-order tests on the group’s main ATLAS page are only a semi-presentation. The identification with the named sporadic simple group is the cited presentation result.

References

ATLAS, Held group, order heading.
ATLAS, full presentation HeG1-P1, displayed relation list and Magma source.